LIDS-P-1614
OCTOBER 1986
FEEDBACK SYSTEM DESIGN WITH AN UNCERTAIN PLANT
D. Milich. L. Valavani, and M.Athans
Laboratorg for Information and Decision SUstems
Massachusetts Institute of Technology
Cambridge, MA 02139 U.S.A.
The mathematical description of a physical plant
(i.e. the nominal model) is always characterized by
The plant/model
uncertainty, or modeling error.
mismatch caused by neglecting high-frequency
phenomena (i.e. unmodeled dynamics) is known as
unstructured uncertainty because only a frequency
dependent magnitude bound on the error is available.
Differences between the actual values and the nominal
values of the parameters in the finite-dimensional,
low-frequency model are the source of structured
uncertainty. .The goal of the robust design method
presented here isto maintain closed-loop stability and
performance in the presence of both types of
uncertainty,
The most popular modern design methodologies
(LQG/LTR, H°) only deal with plant models which
contain unstructured uncertainty [1,21. The effect of
plant parameter variations may be incorporated into the
unstructured uncertainty; however, the directional
(phase) information associated with the structured
uncertainty is lost and the result is an overly
conservative design. Although these methodologies
stability,
closed-loop
nominal
guarantee
stability-robustness with respect to the unstructured
uncertainty, and nominal performance, the Issue of
robust performance is ignored.
Techniques which deal directly with structured
uncertainty have been developed. Horowitz [3-51 has
proposed the so-called Horowitz templates which,
represent, at a particular frequency, the gain and phase
changes associated with parameter variations as a
region on a Nichols chart. A loop transfer function is
derived from the templates which ensures closed-loop
stability and a certain amount of performance over the
possible range of parameters. This procedure is
extremely tedious for more than a few frequency points
and a great deal of judgment is required on the part of
the designer.
Sideris and Safonov [61 approach the problem of
structured uncertainty by examining a plant template in
the complex plane. A series of transformations at each
frequency is performed which maps the irregularly
shaped region in the complex plane onto the unit disk.
The directional properties of the uncertainty are
eliminated and the transformed problem is essentially a
design with unstructured uncertainty. Nevanlinna-Pick
interpolation is used to find the compensator which is
then transformed back in such a way that itis a
solution to the original problem. This approach may be
promising, however it appears that a great deal of
computation is required at each frequency point.
The problem of robust stabilization in the presence
Of parameter uncertainty is addressed by Khargonekar
and Tannenbaum [7]. The authors do not deal with
performance issues directly and simultaneous
variations in the poles and zeros cannot be considered
Doyle [8-101 has
within the present framework.
developed a new mathematical quantity j, the
structured singular value, to handle structured
This research was supported by the NASA Ames and
Langley Research Centers under grant NASA/NAG 2-297,
by the Office of Naval' Research under contract
ONR/N0004-82-K-0582 (NR 606-003), and by the
grant
under
Foundation
Science
National
NSF/ECS-82 10960.
While it may be a valuable
robust performance.
analysis tool [11,121], a feedback synthesis methodology
based on ji is not yet available.
It is the purpose of this paper to develop techniques
for analyzing the stability and performance of a SISO
feedback system which contains a plant model with
Abstract
A method is developed to design a fixed-parameter
compensator for a linear, time-invariant, SISO plant
model characterized by significant structured, as well
as unstructured, uncertainty. The controller minimizes
the H°° norm of the worst-case sensitivity function
over the operating band and the resulting feedback
system exhibits robust stability and robust
It is conjectured that such a robust
performance.
nonadaptive control design technique can be used
on-line in an adaptive control system.
1. Introduction
uncertainty and the problems of robust stability and
gt(s)
parameter uncertainty and unmodeled dynamics. Based
gs,[
1
(s,
(4)
+ A(s)
on the analysis, a robust control design method which
° theory is outlined.
uses concepts from LQG/LTR and H"
The compensator guarantees nominal closed-loop
stability and stability-robustness with respect to
structured and unstructured uncertainty. In addition,
the H°° norm of the worst-case sensitivity function
over the operating band is minimized. Since this norm
provides an upper bound on all possible sensitivity
functions, the closed-loop system satisfies the
condition of robust performance.
In order to design the compensator k(s) and to
analyze the feedback system containing the true plant,
information about the nominal model g(s,~), the
Information about the nom a mode g(s, the
structured uncertainty (s ,Q), and the unmodeled
dynamics A(s)- is required. The nominal model and the
magnitude bound on the unstructured uncertainty AO(M )
magnitude
bound on the unstructured uncertainty
are known a priori. The exact value of S(s,e- ,_) is
unknown, but the set containing all possible values may
be constructed from 8. Using Equation (3) and
replacing ¢_with e. (s,._ may be computed for a
2. Problem Formulation
given-.
Consider the feedback system in Figure 1 with plant
S(s,Qe. = [g(s,D/g(s,)] - 1 for
model g(s) and compensator k(s). The physical plant to
be controlled is single-input, single-output, linear, and
time-invariant. The transfer function gt(s). which may
be of infinite order, fully describes the true plant. The
low-frequency behavior of gt(s) is captured in a
parameterized nth order transfer function g(sq,), where
a represents the parameter vector. The actual value of
is. e. Unstructured uncertainty results from the
fact that the nth order transfer function g(s,- ) cannot
completely describe the (possibly) infinite-dimensional
plant gt(s). Using the multiplicative form of the
modeling error as in [1, the frequency-domain
description of the unstructured uncertainty is
represented by the stable transfer function [I+A(s)].
The true plant can be described as folloWS.
gt(s) = g(s,'
)[ I + A(s)
.6
e
(5)
The above equation defines the set of structured
uncertainty Us.
U 5 (s,- ) = {( (s,eQ
I Qee } for 6 E e
(6)
Us and the set of transfer functions A(s) satisfying
Equation (2) lead to the definition of the set of possible
plant transfer functions G(s) which contains the true
plant gt(s).
G(s) = ((s)S (s,,)EUs(s. ) , A(j) _< A0(s) }(7)
where g(s) = g(s,X[l + S(s,.J)[1 + A(s)l.
(I)
In any practical situation _*(
is unknown; however,
the true parameter values are bounded by the known set
f8 (i.e. &( e 0). For design purposes a nominal
parameter vector 6 is chosen from the set e.
Structured uncertainty,. which arises when A and Q are
not identical, can be described In tne frequency-domain
using the multiplicative model error representation as
in Equation (I ).
=
g(s,e")
g(s,)[ I + (s, . _ ]
(3)
The design method in section 4 will find a
compensator k(s) such that the closed-loop system is
stable for all g(s) e G(s). In addition to robust
stabilization, the compensator must achieve closed-loop
performance and performance-robustness.
The
sensitivity transfer function s(s) = [l+g(s)k(s)I-'
evaluated
along
the
jo-axis governs the
command-following
and
disturbance-rejection
performance of the feedback system in Figure 1. The
performance objective of the design method is to
minimize the H' norm of the worst-case sensitivity
function over the frequency range where the commands
and disturbances have energy (i.e. the operating band
Qo), subject to closed-loop stability. By definition, the
magnitude of the worst-case sensitivity function is an
upper bound on the magnitude of the sensitivity
From (1) and (3), the true plant gt(s) can be represented
in terms of the low-frequency model, the parametric
uncertainty, and the unmodeled dynamics.
feedback system satisfies the condition of robust
performance with respect to the worst-case
sensitivity.
It is assumed that a frequency-dependent magnitude
bound on the unstructured uncertainty is available, but
the phase of A(jw) is completely unknown.
function for all g(s) E G(s) and for all A.
2
Thus the
...
.-..
-..
·.. .....
3. Analusis
uncertainty to change the number of critical point
encirclements. From Figure 2, the distance to the
This section covers the stability and performance
analysis of the feedback system in Figure 1. The
compensator k(s) has been designed based on a plant
g(s) e G(s). The loop transfer function t(s)=g(s)k(s) and
the sensitivity s(s) are the functions of interest. Let
t(s,f) = g(s4)k(s) be the nominal loop transfer function.
Then,
critical point dc of the loop t(jw) may be computed.
The closed-loop system may be unstable if the critical
point is located in the interior of the circular region in
Figure 2 (i.e. dc is negative).
t(s) = t(s,[ 1 + s(s,e.) ] I + a(s) I
= t(s,~ + t(s,)(s.
~. ~~~..
(8a)
.) + t(s,)[l++S(s.~)]A(s)
I +t(jO
~.
A)[l+8(ja).,]l
I t(jt,4,[I +$
s(j,,.9~]
A(jo) i
(9)
where 8(jc,-Q.) e Us(j.), IA(jo)
|
|! ao
0 ()
(8b)
To perform a worst-case stability analysis the
minimum value of dc(oX.) must be found by searching
3.1 Stability
At the very least, the compensator k(s) ensures the
stability of the nominal closed-loop system. From the
Nyquist criterion the number of encirclements of the
critical point in the Nyquist diagram of t(jio,~) is
known. At this point it must be assumed that g(sQ)
has the same number of unstable poles for all -. e 8.
Since the structured uncertaintu does not change the
number of unstable poles and A(s) is stable, the Nyquist
criterion requires the number of encirclements of the
critical point to remain unchanged for the loop transfer
function tp(j).
A stability-robustness condition may be derived by
examining the loop transfer function t(jo) in the
complex plane at a particular frequency X (Figure 2).
From Equation (8b), there are two perturbation terms
added to t(jw,) which may alter the stability of the
nominal loop (i.e. change the number of encirclements
of the critical point). The perturbation caused by the
structured uncertainty is t(j),s)&(jw,.A)
and the
second term, t(jwo[l-sc(jw,&, Wjw), is a result of
the combined effect of structured and unstructured
uncertainty.
This perturbation is represented as a
circle in the complex plane because the phase of A(jco)
is unknown.
Before a stability-robustness test can be derived, it
must be assumed that the structured uncertainty does
not alter the stability of the nominal loop (i.e. the
perturbation t(jw,$)S(j&o,.Q
does not change the
number of critical point encirclements of the nominal
loop t(jw,) ). Note that this is a very restrictive
assumption; however, it is only used to derive a
stability-robustness condition for design purposes and
it does not limit the classes of uncertainty which may
be considered. Under this assumption the loop transfer
function t(jw) will be closed-loop stable if the
distance to the critical point ( i.e. Il+t(jw)
dc(w.QA)
) is
greater than zero over all frequencies. If this is the
case it is impossible for the perturbations due to
over the set Us(jzw.) and by replacing A(jw)l
by its
bound
8 c(,)=
min
l+t(jw,)[l+S(jo,Q
I{
seUs
It(jc
I-)]
})[I+(jo
(IO)
Theorem 1: Sufficient Condition for Robust Stability
for all w, then the true closed-loop
system is stable.
EEf
From Figure 2 and the above assumption,
the number of encirclements of the critical
point--cannot-be-altered-by the-uncertainty
if the inequality in Theorem 1 is satisfied.
Note the fundamental difference between structured
and unstructured uncertainty. In the case of structured
uncertainty, directional
(phase)
information
is
exploited. That is, only structured perturbations which
decrease the distance to the critical point are of
concern, and perturbations away from the critical point
increase stability-robustness and can be ignored. No
phase information is available for the unstructured
uncertainty. Therefore it must be assumed that the
unstructured perturbations are always in the direction
Of the critical point.
3.2 Performance
From the definition oc the sensitivite fution,
s(s)=l+t(s) - l , performance
interpreted from the
Nyquist diagram in terms of the distance to the critical
point. That is,
Js(ji)j = /dc(oe,~)
(ti)
The worst-case sensitivity function magnitude is found
as an immediate consequence of Equation -( 1).
Is'(jwi)l = 1/a'c(W,)
(12)
Define a scalar measure of performance E as the °H°
norm of the worst-case sensitivity over the operating
E=
max i
,EQ
'(V)i
s(13)
The objective of the robust design method
minimize E (i.e. maximize performance), subject
closed-loop stability condition in Theorem 1.
s(jw) < s(jw)I for all g(s) EG(s) and for all
requirement of robust performance is satisfied.
is to
to the
Since
, the
4. Robust Design Method
A method is proposed to-design a compensator whictrh
guarantees closed-loop stability and performance in the
presence of uncertainty (i.e. robust stability and robust
performance). The controller minimizes the Ho norm of
the worst-case sensitivity function over the operating
band.
The concept of Nuquist-shaping is introduced as an
integral part of the robust design process. While
loop-shaping techniques are concerned with tailoring
the magnitude of the loop transfer function,
Nyquist-shaping requires magnitude and phase
information to construct a target Nuquist diagram. The
target satisfies all stability and performance
conditions and represents the desired nominal loop
transfer function. For the given plant, the robust
design method produces a compensator which yields a
nominal loop transfer function t(jwi) whose Nyquist
diagram approaches the target to any required degree of
accuracy.
It is assumed that the low-frequency, nominal plant
model g(s,), the parameter set 8 (and hence Us), and
the magnitude bound on the unstructured uncertainty
a(Uo) are known to the control system designer. Then,
the first step in the robust design method consists of
finding the target Nyquist diagram via the
Nyquist-shaping procedure. The target curve cannot
have an arbitrary shape in the complex plane. Stability
and analytic function theory require the target Nyquist
diagram to satisfy the following constraints.
(1) Nyquist stability criterion
(2) Bode's Integral Theorem [131
Theorem
131
(2) Bode's Integral
(3) Theorem 1
Subject to the above constraints, the target must
minimize the H°° norm of the worst-case sensitivity
The task of
function over the operating band.
incorporating these constraints into a standard
optimization problem is a subject of current research.
While not all of the details of the Nyquist-shaping
algorithm have been rigorously formalized, the basic
idea is to start with a Nyquist diagram which
corresponds to a loop transfer function which is known
to be closed-loop stable with respect to the structured
° and unstructured uncertainty in the given plant model
(i.e. satisfies Theorem 1). This initial curve in the
complex plane is continuously deformed in such a way
that the above constraints are met and the HOO norm of
the worst-case sensitivity function over X E Q0 is
monotonically decreasing. The procedure terminates
when acceptable performance has been achieved, or
when it is no longer possible to improve worst-case
performance without violating the stability-robustness
condition in Theorem 1.
Finding the initial curve for the Nyquist-shaping
algorithm is relatively straight-forward. Since current
methodologies can handle unstructured uncertainty, the
idea is to transform the original problem (with
structured and unstructured uncertainty) Into one with
just unstructured uncertainty. This must be done in a
conservative manner so that the resulting compensator
guarantees a stable closed-loop system.
A new multiplicative uncertainty may be defined by
lumping the structured and unstructured .uncertainty
together.
'
=
(s,_,)
(s,e.
+ [i+$s(s,a.)A(s)
(14)
The plant model g(s) is represented as follows.
g(s) = g(s,)[lI + A'(s,8)
(15)
]
By ignoring the directional properties of the structured
uncertainty a frequency-dependent magnitude bound on
A'(s,j.)is computed at each frequency.
A
A'o(&,4) = max {j 8(jI.a) | + |1 +(j,
o()
'(
, for a w
) for
) } (16)
(17)
LQG/LTR t 41 or
the unstructured uncertainty od
the unstructured uncertainty A'O(,@, LOG/LTR [141 or
another suitable control design methodology is used to
find a compensator ko(s) which guarantees a stable
closed-loop system. The Nyquist diagram of the loop
transfer function to(jo)=g(jo,j)ko(jo) is the startLng
point for the Nyquist-shaping procedure. Note that this
is an overly conservative control system design because
the directional nature of the structured uncertainty has
not been used. The Nyquist-shaping procedure results
in a target Nyquist diagram which should remove this
conservatism and improve performance.
The target curve will typically correspond to a
high-order
system.
A
finite-dimensional,
parameterized loop transfer function must be obtained
from the target Nyquist diagram.
This can be
accomplished by a least-squares fit to the magnitude
and phase data at specific frequency points.
The
stability of the finite-dimensional target should be
checked by Theorem 1. If the robustness condition is
not satisfied, a frequency-weighted least-squares
procedure is used to improve the transfer function fit
in the frequency range where Theorem 1 was violated.
Alternatively, a higher order transfer function can be
used to fit to the target Nyquist diagram.
The finite-dimensional target loop is used in the
Formal Loop Shaping LQG/LTR methodology to arrive at
a compensator k(s). In [151, Stein and Athans outline
the framework for using LQG/LTR to recover arbitrary
(stable, minimum phase) target loop transfer functions.
That is, the LQG/LTR methodology can be used to find a
compensator k(s) such that the magnitude of the loop
.transfer function g(s,)k(s) matches the magnitude of
the target loop.
Unfortunately, this application of
LQG/LTR with Formal Loop Shaping requires the plant
model g(s,_) to be stable and minimum phase. Research
is being--conducted to remove-this-restriction.
given to an issue at the heart of the adaptive control
problem: what are the Derformance benefits of adaptive
control? In theory an adaptive control system provides
better performance with respect to a fixed-parameter
compensator because more information about the
physical plant is incorporated into the design process
(on-line).
However, robust adaptive control systems rely upon
some combination of external persistently exciting
signals (to ensure good identification), slow sampling
(to provide stability robustness with respect to
unmodeled high-frequency dynamics, [211), and extensive
real-time computation (to provide safety nets which
turn off the adaptation when it exhibits instability,
[221).
These
robustifying measures
degrade
command-following
and
disturbance-rejection
performance and tend to neutralize the anticipated
benefits of an adaptive compensator. In light of these
circumstances it is imperative that the decision to use
adaptive control, for a real engineering application, is
based upon a quantitative assessment of the costs and
the benefits of an adaptive system. The robust.design
method proposed here produces the nonadaptive
feedback system which minimizes the H° norm of the
worst-case sensitivity function over the operating
band.
This system may serve as a performance
benchmark to which an adaptive control system is
compared.
6. C
5. imDlications For Adaptive Control
A frequency-domain analysis of the stability and
performance of a SISO feedback system with structured
and unstructured uncertainty has been performed. The
crucial analysis parameter is the distance to the
critical point in the Nyquist diagram. Directional
information (inthe complex plane) associated with the
structured uncertainty
is exploited to reduce
conservatism.
A new method is outlined to design
linear, time-invariant compensators for $10S
plant
models characterized by parameter uncertainty and
unmodeled dynamics. The resulting feedback system
minimizes the H0° norm of the worst-case sensitivity
function over the operating band and is guaranteed to be
closed-loop stable. The concept of Nyquist-shaping was
introduced as an integral part of the design process.
It is conjectured that the robust design method can
be used on-line in an adaptive control context. However
before the decision to use adaptive control is made, the
control designer must have a quantitative measure of
the performance
improvement over the 'best"
nonadaptive system. The robust design method provides
a benchmark for the performance comparison.
A very active search for a robust adaptive control
methodology is being conducted and a trend in the
literature is developing. Many researchers now believe
that a robust adaptive control system must consist of a
robust system identification algorithm coupled with a
robust control design method [16-201. This philosophy
has been referred to as adaptive robust control, in
contrast to robust adaptive control.
Compensator
redesign takes place infrequently compared to the
system sample period and only when more accurate
information about the system can be provided by the
identification algorithm.
While the robust design
method presented here will be useful in its own right
for fixed-parameter compensator design, the goal is to
develop an on-line algorithm as part of a practical (i.e.
robust) adaptive control system. In addition, the design
method will provide the initial guess for the adaptive
compensator.
Over the years a great deal of attention has been
paid to the development of specific adaptive
algorithms; however, very little consideration has been
5
[151
Acknowledgments
The authors wish to thank Rick LaMaire and Dan
Grunberg for their valuable editorial assistance.
[161
[17]
[181
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[191
[11
[21
[31
[41
[51
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.....
r(s)
d(s)
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(us
(ss)
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Figure I. Feedback System.
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-
\d
(.)
c
t(jo,
)
t( j
8( jcL)
t(j8,_)t+s(jW,eQ)]A(jo)
Figure 2.
Representation of.t(jo)
in the complex plane.
6
real